A190272 Numbers n such that n' = a -1, with n and a semiprimes and gcd(a,n) > 1, where n' is the arithmetic derivative of n.

6, 14, 15, 22, 33, 38, 46, 51, 62, 86, 87, 91, 95, 118, 141, 142, 145, 158, 159, 166, 206, 249, 262, 267, 278, 287, 295, 321, 326, 382, 395, 398, 411, 422, 445, 446, 473, 502, 519, 537, 542, 545, 581, 591, 622, 662, 695, 699, 703, 718, 745, 758, 766, 789, 838, 886, 895, 926, 951, 958, 995, 998, 1046, 1126, 1139, 1145, 1167, 1199, 1238, 1262, 1318, 1329, 1347, 1382, 1401, 1441, 1486, 1678, 1707, 1717, 1718, 1726, 1745, 1757, 1761, 1766

Offset

1,1

Comments

This sequence is infinite, assuming Dickson's conjecture. In fact, the conjecture implies that there are infinitely many terms of this sequence divisible by any fixed prime p. - Charles R Greathouse IV, May 08 2011

Related to the Rassias Conjecture ("for any odd prime p there are primes q < r such that p*q = q+r+1") setting n = q*r, a = q+r+1. The sequence includes the cases with p = q (or p = r). Generalization can be achieved by removing the semiprimality condition or accepting gcd(n,a)=1. The differential equation in its general form n' = a + 1 includes Primary Pseudoperfect numbers, i.e., n' = n-1 (A054377).

Links

Table of n, a(n) for n=1..86.

For Rassias conjecture: Preda Mihăilescu, Review of Problem Solving and Selected Topics in Number Theory, Newsletter of the European Mathematical Society, March 2011, p. 46.

Formula

Semiprimes pq with (p+q+1)/p prime. - Charles R Greathouse IV, May 08 2011

Example

For n=6, 6' = 5, a = 5-1 = 4, gcd(4,6)=2, so 6 is a term.

Maple

der:=n->n*add(op(2, p)/op(1, p), p=ifactors(n)[2]);
# for quick reference only
seq(`if`(bigomega(i)=2 and bigomega(der(i)+1)=2 and gcd(i, der(i)+1)>1, i, NULL), i=1..2000);

Prog

(PARI) find(lim)=my(v=List()); forprime(p=2, sqrtint(lim\2), forstep(q=2*p-1, lim\p, p+p, if(isprime(q\p+2)&isprime(q), listput(v, p*q)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, May 08 2011

Crossrefs

Cf. A001358 (semiprimes), A003415 (arithmetic derivative), A054377 (Primary Pseudoperfect).

Sequence in context: A230873 A393065 A192321 * A107982 A341448 A081535

Adjacent sequences: A190269 A190270 A190271 * A190273 A190274 A190275

Keyword

nonn

Author

Giorgio Balzarotti, May 07 2011

Status

approved